Quantification of market efficiency based on informational-entropy

Abstract: Since the 1960s, the question whether markets are efficient or not is controversially discussed. One reason for the difficulty to overcome the controversy is the lack of a universal, but also precise, quantitative definition of efficiency that is able to graduate between different states of efficiency. The main purpose of this article is to fill this gap by developing a measure for the efficiency of markets that fulfill all the stated requirements. It is shown that the new definition of efficiency, based on informational-entropy, is equivalent to the two most used definitions of efficiency from Fama and Jensen. The new measure therefore enables steps to settle the dispute over the state of efficiency in markets. Moreover, it is shown that inefficiency in a market can either arise from the possibility to use information to predict an event with higher than chance level, or can emerge from wrong pricing/ quotes that do not reflect the right probabilities of possible events. Finally, the calculation of efficiency is demonstrated on a simple game (of coin tossing), to show how one could exactly quantify the efficiency in any market-like system, if all probabilities are known.